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The field of elimination theory was motivated by the need of methods for solving systems of polynomial equations.. One of the first results was Bézout's theorem, which bounds the number of solutions (in the case of two polynomials in two variables at Bézout time).
Animation of Gaussian elimination. Red row eliminates the following rows, green rows change their order. In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients.
In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors , direct methods would deliver an exact solution (for example, solving a linear system of equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } by Gaussian elimination ).
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
LU decomposition on Math-Linux. LU decomposition at Holistic Numerical Methods Institute; LU matrix factorization. MATLAB reference. Computer code. LAPACK is a collection of FORTRAN subroutines for solving dense linear algebra problems; ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, etc. C++ code, Prof. J. Loomis, University ...
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.
Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier [ 1 ] who proposed the method in 1826 and Theodore Motzkin who re-discovered it in 1936.
The main theorem of elimination theory is a corollary and a generalization of Macaulay's theory of multivariate resultant. The resultant of n homogeneous polynomials in n variables is the value of a polynomial function of the coefficients, which takes the value zero if and only if the polynomials have a common non-trivial zero over some field ...
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